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On the fixed locus of the antisymplectic involution of an EPW cube

Francesca Rizzo

TL;DR

The paper determines the fixed locus $\mathsf{W}_A$ of the antisymplectic involution on EPW cubes, proving it is a rigid atomic Lagrangian threefold with canonical bundle $\omega_{\mathsf{W}_A}=\mathscr{O}_{\mathsf{W}_A}(2)$ and Euler characteristic $\chi_{top}(\mathsf{W}_A)=-1200$. The authors develop a framework combining LLV decomposition for very general HK manifolds of $K3^{[3]}$-type, detailed analysis of Hodge classes and automorphism actions, and a degeneration strategy in the style of Flappan–Macrì–O'Grady–Saccà to control the fixed locus in a degenerating family. They explicitly compute the cohomology class of $\mathsf{W}_A$ in $H^6(\widetilde{\mathsf{Z}}_A,\mathbb{Z})$ as $[\mathsf{W}_A]=\frac{5}{8}(3h^3-hc_2)$ and derive Chern data for $\mathsf{W}_A$, including $c_1$, $c_2$, and $c_3$. A singular degeneration to a central fiber $X$ is constructed, revealing the fixed locus as a union of components with a detailed local description, and the degeneration proves the rigidity of $\mathsf{W}_A$ by relating its cohomology to that of the central fiber $F_3$ via a flat family.

Abstract

EPW cubes are polarized hyper-Kähler varieties of K$3^{[3]}$-type that carry an anti-symplectic involution. We study the geometry of the fixed locus $\sW_A$ of this involution and prove that it is a \emph{rigid} atomic Lagrangian submanifold. Our proof is based on a detailed description of certain singular degenerations of EPW cubes and the degeneration methods of Flappan--Macrì--O'Grady--Saccà.

On the fixed locus of the antisymplectic involution of an EPW cube

TL;DR

The paper determines the fixed locus of the antisymplectic involution on EPW cubes, proving it is a rigid atomic Lagrangian threefold with canonical bundle and Euler characteristic . The authors develop a framework combining LLV decomposition for very general HK manifolds of -type, detailed analysis of Hodge classes and automorphism actions, and a degeneration strategy in the style of Flappan–Macrì–O'Grady–Saccà to control the fixed locus in a degenerating family. They explicitly compute the cohomology class of in as and derive Chern data for , including , , and . A singular degeneration to a central fiber is constructed, revealing the fixed locus as a union of components with a detailed local description, and the degeneration proves the rigidity of by relating its cohomology to that of the central fiber via a flat family.

Abstract

EPW cubes are polarized hyper-Kähler varieties of K-type that carry an anti-symplectic involution. We study the geometry of the fixed locus of this involution and prove that it is a \emph{rigid} atomic Lagrangian submanifold. Our proof is based on a detailed description of certain singular degenerations of EPW cubes and the degeneration methods of Flappan--Macrì--O'Grady--Saccà.

Paper Structure

This paper contains 15 sections, 20 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Hyper-Kähler manifold s of $\mathop{\mathrm{K3}}\nolimits^{[3]}$-type
  3. Hodge classes on very general hyper-Kähler manifolds of K$3^{[3]}$-type
  4. Action of automorphisms on the LLV algebra
  5. Hodge classes on very general polarized hyper-Kähler manifolds of K$3^{[3]}$-type
  6. EPW cubes
  7. Euler characteristic of ${\mathsf{W}}_A$
  8. The class $[{\mathsf{W}}_A]$ in $H^6(\widetilde{Z}_A, {\bf Z})$
  9. Chern classes
  10. A singular degeneration of $\widetilde{{\mathsf{Z}}}^{\ge 2}_A$
  11. The special fiber of $\mathscr{X} \to D$
  12. Fixed locus on the special fiber
  13. Local description of $X$
  14. Degeneration of ${\mathsf{W}}_A$
  15. Cohomology groups $H^p(F_3, \mathscr{O}_{F_3})$

Key Result

Theorem 1.1

Let $\widetilde{{\mathsf{Z}}}_A$ be a smooth EPW cube with associated involution $\iota_A$. The fixed locus ${\mathsf{W}}_A$ of $\iota_A$ is a smooth irreducible threefold of general type, with canonical bundle $\omega_{{\mathsf{W}}_A}=\mathscr{O}_{{\mathsf{W}}_A}(2)$ and self-intersection number in

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 29 more